On residually finite knot groups

Mayland, E. J.

doi:10.1090/S0002-9947-1972-0295329-1

On residually finite knot groups
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Trans. Amer. Math. Soc. 168 (1972), 221-232 Request permission

Abstract:

The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag’s results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surface has complement with free fundamental group. As a by-product more explicit finite representations, namely cyclic extensions of certain $p$-groups, are obtained for these knot groups and the groups of fibred knots. Finally composites of two such knots also have residually finite groups.

References

Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. I. The groups, Trans. Amer. Math. Soc. 129 (1967), 308–321. MR 217157, DOI 10.1090/S0002-9947-1967-0217157-3
Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. II. Properties, Trans. Amer. Math. Soc. 142 (1969), 507–538. MR 245653, DOI 10.1090/S0002-9947-1969-0245653-3
D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. MR 192475, DOI 10.1112/plms/s3-16.1.369
R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR 87652, DOI 10.1112/plms/s3-7.1.29
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215

Theory of groups

15

18

Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802

Boundary preserving mappings of $3$-manifolds

B. H. Neumann, An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554. MR 62741, DOI 10.1098/rsta.1954.0007
Hanna Neumann, Generalized free products with amalgamated subgroups, Amer. J. Math. 70 (1948), 590–625. MR 26997, DOI 10.2307/2372201
Hanna Neumann, Generalized free products with amalgamated subgroups. II, Amer. J. Math. 71 (1949), 491–540. MR 30522, DOI 10.2307/2372346

Knot theory

Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), no. 3, 57–104 (German). MR 0031733

Lehrbuch der Topologie

Peter Stebe, Residual finiteness of a class of knot groups, Comm. Pure Appl. Math. 21 (1968), 563–583. MR 237621, DOI 10.1002/cpa.3160210605
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594

On doubled knots

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